What is method of variation of parameters?

Overview

The method of variation of parameters is the general method which we use to find out a particular solution of a differential equation by replacing the constants in the solution of the homogeneous differential equation by functions and evaluating these functions so that the original DE will be satisfied.

Method of variation of parameters

Consider a second order LDE with constant coefficients given by

Then let the complementary function is given by

Then the particular integral is

Where u and v are unknown and to be calculated using the formula

Solved examples

Example-1: Solve the following DE by using variation of parameters-

Sol. We can write the given equation in symbolic form as-

To find CF-

It’s AE is

So that CF is –

To find PI-

Here

Now

Thus PI will be

So that the complete solution is-

Example-2: Solve the following by using the method of variation of parameters.

Sol. This can be written as-

C.F.-

Auxiliary equation is-

So that the C.F. will be-

P.I.-

Here-

Now 

Thus 

So that the complete solution is-

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